## How much is an interest rate reduction worth?

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DoingHomework
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### How much is an interest rate reduction worth?

Ok, I'm good with the math but I'm having trouble thinking through the logic here:

Suppose I have a mortgage at 6% that I owe \$100000 on. Suppose I can refinance it for \$0 and still owe \$100000 but at 4%. How much is that worth? That is, suppose I had to pay \$xxx to make the change in interest rate, what is the maximum \$xxx should be?

stannius
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### Re: How much is an interest rate reduction worth?

It depends (among other things) on how long you expect the rate to be in effect.

DoingHomework
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### Re: How much is an interest rate reduction worth?

Yes, of course it depends on term but assume they are equal for the two options.

Here's my problem - if someone asked me the question I would say that one just computed the net present value (NPV) for each option and then compares. But the NPV for each of these is \$100,000!

I know there has to be a simple approach but I'm having trouble thinking through what it is. Thinking about it since yesterday I now think that the correct approach is to find the net present value of the difference in payments over the term. So, in this case I get:

Payment at 4% is \$5783.01per year
Payment at 6% is \$7264.89 per year

If I compute the NPV of the difference using 4% I get \$25624.74

That means I should be willing to pay \$25624.74 to buy the rate down from 6% to 4%, right?

bel
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### Re: How much is an interest rate reduction worth?

This is how I would look at it... A 30-yr mortgage at 6% interest requires \$600 (rounded up) monthly payment. If I pay the same amount on a 30-yr mortgage with 4% interest, the 4% mortgage will be paid off in 20 yrs 4 mos.

So 20 yrs 4 mos in the future,

balance on 4% mortgage = \$0
balance on 6% mortgage = \$52,675

Then I ask myself, how much do I need to have today to apply on the 6% mortgage to have it paid off in 20 yrs 4 mos? Answer = \$15,600

So, if I have \$15,600 today, I can choose to use that to pay for a refinance to get a 4% mortgage OR apply it towards the 6% mortgage. Either choice will result to a paid off mortgage in 20 yrs 4 mos.

How does that sound?

DoingHomework
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### Re: How much is an interest rate reduction worth?

bel wrote:... how much do I need to have today to apply on the 6% mortgage to have it paid off in 20 yrs 4 mos? Answer = \$15,600
How does that sound?

Makes sense. But you are basically discounting at 6% rather than 4%. If I do that with my calculation I get a similar result to you. The difference probably comes from roundng though I did not check.

I'd love to hear what others say as well.

This is the calculation that needs to be done when considering paying points on a mortgage, which I am considering right now. I am being offered about 12+ choices of rates, points, and required LTV on a refinance. The highest rate is 4.25% and they go down from there by about half a % depending on points I am willing to pay. The thing is, when I apply this calculation I find that the cost in points of lowering the rate is about half the value which means it is the best deal to pay the points. That does not necessarily make sense to me but that is what I am being offered.

bel
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### Re: How much is an interest rate reduction worth?

Yes, considering the new case, \$xxx = \$25,582. This will give a different choice between:

6% mortgage of \$100k
4% mortgage of \$100k + \$25,582

My original calculation considered an existing 30-yr mortgage with a choice of refinancing for \$xxx or not, as indicated in the original post.

bel
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### Re: How much is an interest rate reduction worth?

Hmmm... Thinking about it further, it depends if \$xxx is paid today or added into the loan amount. If you have \$xxx and use it today, then \$xxx = \$15k, otherwise it's \$25k. I think that's right.

stannius
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### Re: How much is an interest rate reduction worth?

DoingHomework wrote:Yes, of course it depends on term but assume they are equal for the two options.

Will either loan be kept until full term? And, are you sure?

DoingHomework wrote:Makes sense. But you are basically discounting at 6% rather than 4%. If I do that with my calculation I get a similar result to you. The difference probably comes from roundng though I did not check.

I had typed part of a response but my baby ate it. But it had started with the assumption that interest rate and discount rate are one and the same. It is common to assume that, but sometimes it leads to odd results like yours where two clearly different loans have the same NPV.

The PV of a series of \$600 payments at 4% discount rate = \$125,582.60
The PV of a series of \$477 payments at 6% discount rate = \$79,628.87

You could also discount the payments at the inflation rate. That measures what a dollar tomorrow will be worth, right? In that case the series of payments on the 6% loan is worth \$133,517 today; the 4% loan payments are worth \$106,318; the difference being \$27,199.

So, based on that I might pay between \$20,371 and \$27,199 in refinancing costs.

I have never heard of a loan with 20+ points, so like you say, it seems like it's always worth paying points. I guess most people either don't have the cash to spare, or plan to move/refinance/etc before up-front points would pay off.

DoingHomework
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### Re: How much is an interest rate reduction worth?

Thanks Stannius and Bel,

The real situation is not exactly what I represented. I was trying to simplify things. And really, I am trying to figure out how to value options for a new loan for comparison. That means that factors like how long I will have the loan and so forth shouldn't matter because they will be the same for each alternative. I'm going to try to clarify here:

I currently have a 5.99% mortgage with 28 years left. The balance is \$340,000. I can refinance with any of the following choices:

4% APR with 0.75 points
4.125% with 0.125 points
4.25% with -0.625 points (i. e. they would rebate)

If the value of the property is lower than we expect an alternative might be

4.00%,1.5 points
4.125%, 0.875 points
4.250%, 0.125 points

Rate could be as low as 3.75% in another situation with 2 points and a low LTV.

We are in a position to apply cash to meet some of the LTV targets. I intend to value the cash payments at the rate I would otherwise earn on the cash. I would take money from a bond fund that I expect to get 5.5% from. So, if we need to put \$10000 cash in to get the LTV right to qualify for 4%, 1.5% for example, I would factor that in as a \$550 cash flow we are giving up in perpetuity...or at least for 30 years. The point is, I can value that part of it.

I am having real difficulty thinking through how I value 4%, 1.5 pts vs 4.25%, 0.125 pts for example. A loan at 4% for 30 years has payment of \$477.42/\$100k. 4.25% has payment of \$491.94/\$100k. The PV of each at their respective interest rates is \$100,000 of course.

I could value each at a different interest rate and compare the difference in PV to the immediate cost in points. That's how I think it should be done. But what rate do I use? If my choices are 4% or 4.25% then it does not make much sense to me to use my current rate of 5.99% or my expected portfolio return or anything else like that.

Like I said, I can do the math! But I'm really having difficulty here figuring out what the right rate to use is for comparing.

DoingHomework
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### Re: How much is an interest rate reduction worth?

Maybe I am answering my own question here but how does this sound?

\$100,000 at 4% has a payment of \$477.42 a month.

\$100,000 at 4.25% has a payment of \$491.94 a month.

The difference is \$14.52 a month. So the value of lowering the interest rate from 4.25% to 4.00% should be the present value of the \$14.52 payment savings over the 30 years. If I use 4% for the interest rate for this calculation I get \$3042.34 for teh value of teh savings. Using 4.25% values the savings at \$2952.51.

Since in my case I am being offered this savings at a price of 1.375 points or \$1375, it seems like it is a great deal.

Have I missed something or done something wrong?

stannius
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### Re: How much is an interest rate reduction worth?

DoingHomework wrote:That means that factors like how long I will have the loan and so forth shouldn't matter because they will be the same for each alternative.

You keep saying that but it's not true. Estimating how long the loan will be in effect is essential to this calculation. The reason is right in your own math - you calculated the PV of a payment of 14.52 over 30 years. If the loan is not in effect for the full thirty years, you don't "receive" all the payments.

Imagine an extreme example. The day after your refinance is complete, you decide to pay off the loan. Maybe you won the lottery or sold the house or whatever. Regardless, if you chose a 4% loan you have had it for one day, but paid 1.375 points for the rate - your total cost for one day is \$10.95 interest plus \$1375 in points, total \$1386. If you chose a 4.25% loan your cost for one day is \$11.65.

If you are saying "I plan to keep this loan for the full 30 years," that's fine. But you can't just say "the term of the loan will be the same either way" and use that to hand-wave away this important consideration.

bel
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### Re: How much is an interest rate reduction worth?

There's definitely a lot of variables. So I'll make some assumptions.

Let,

Person A has the 4% mortgage
Person B has the 4.25% mortgage

If I am person B, how much \$xxx would I be willing to pay A for his 4% mortgage?

Option 1:
If B adds \$xxx into the loan, then B wants to pay A less than \$7,760 to make the switch worthwhile. So then, B's new loan will have to be less than \$107,760 at 4% interest.

Option 2:
If B gets the \$xxx from an investment with ROR of 5.5%, \$xxx will be \$4,957. So, B should not pay A more than \$4,957 to get the 4% mortgage.

Right now, this make sense to me. LOL.

DoingHomework
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### Re: How much is an interest rate reduction worth?

stannius wrote:You keep saying that but it's not true. Estimating how long the loan will be in effect is essential to this calculation. The reason is right in your own math - you calculated the PV of a payment of 14.52 over 30 years. If the loan is not in effect for the full thirty years, you don't "receive" all the payments.

Imagine an extreme example. The day after your refinance is complete, you decide to pay off the loan. Maybe you won the lottery or sold the house or whatever. Regardless, if you chose a 4% loan you have had it for one day, but paid 1.375 points for the rate - your total cost for one day is \$10.95 interest plus \$1375 in points, total \$1386. If you chose a 4.25% loan your cost for one day is \$11.65.

If you are saying "I plan to keep this loan for the full 30 years," that's fine. But you can't just say "the term of the loan will be the same either way" and use that to hand-wave away this important consideration.

Ok, I'll buy that. And since our plan is to have it maid off in about 10 years that could make a big difference. I had not looked at it that way.